Description: "A most fundamental
fixed point theorem" of Alexander Abian (1923-1999),
apparently proved in 1998. Let , ,
,... be the iterates of . The
theorem reads (using our variable names): "Let be a mapping from
a set into
itself. Then has a
fixed point if and only if:
There exists an element of such
that for every ordinal
, is an element of , and if is not a
fixed point of
then the 's are all distinct for every
ordinal ." See df-rdg6439 for the operation. The
proof's key idea is to assume that does not have a fixed point,
then use the Axiom of Replacement in the form of f1dmex5767 to derive that
the class of all ordinal numbers exists, contradicting onprc4592. Our
version of this theorem does not require the hypothesis that be a
mapping. Reference:
http://us2.metamath.org:88/abian-themostfixed.html.
For an
application of this theorem, see
http://groups.google.com/group/sci.stat.math/msg/1737ee1133c24aeb
for
its use in a proof of Tarski's fixed point theorem. (Contributed by NM,
5-Sep-2004.) (Revised by David Abernethy,
19-Jun-2012.)